The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

Abstract

We consider the existence of the periodic solutions in the neighbourhood of equilibria for 𝐶^∞ equivariant Hamiltonian vector fields. If the equivariant symmetry 𝑆 acts antisymplectically and 𝑆²=𝐼, we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds each containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian systems.

1. Introduction

We first give some definitions for our problem. A 2𝑛×2𝑛 matrix 𝑇 is called (anti)symplectic if 𝑇^𝑇𝐽𝑇=±𝐽. Consider a 𝐶^∞ vector field 𝑓∶𝑂⊂ℛ^𝑁→ℛ^𝑁 and the system𝑑(𝑥)𝑑𝑡𝑥=𝑓. (1.1) Let 𝑆 be a diffeomorphism of ℛ^𝑁 into itself. If 𝑆𝑓=𝑓𝑆, we call that the system (1.1) is 𝑆-equivariant or the vector field 𝑓 is 𝑆-equivariant. Denote by 𝐼 the identity matrix. In this paper, 𝑆 satisfies 𝑆²=𝐼. When the system (1.1) is 𝑆-equivariant, if 𝑥(𝑡) is a solution, then 𝑆𝑥(𝑡) is also a solution. An orbit 𝑥(𝑡) is called symmetric if it is 𝑆-invariant; that is, 𝑆𝑥(𝑡)=𝑥(𝑡).

Problems. Consider a system 𝑥̇=𝐴𝑥+𝑔(𝑥), 𝑥∈𝑅^𝑛 with 𝐴 being an 𝑛×𝑛 matrix and 𝑔=𝑂(𝑥²) a 𝐶^∞ vector function. Suppose that the system has a nondegenerate integral. Suppose that 𝐴 has a pair of purely imaginary eigenvalues ±𝑖 and no other eigenvalues of the form ±𝑘𝑖, 𝑘∈𝑍. That is, the eigenvalues ±𝑖 are nonresonant with the other ones. Then, the well-known Liapunov Center theorem tells us that there exists a one-parameter family of periodic orbits emanating from the equilibrium point with the period being close to 2𝜋 as they approach to the equilibrium. We call such families the Liapunov Center families.

This result can be used easily to Hamiltonian systems and obtain existence of periodic solutions. Later, many mathematicians were dedicated to study periodic solutions of Hamiltonian systems and tried to generalize the result. Gordon [1] obtained an infinite number of periodic solutions in arbitrarily small neighborhoods of the origin for Hamiltonian systems with convex potential. Weinstein [2] obtained the Liapunov Center families with no eigenvalue assumptions when the equilibrium point is a nondegenerate minimum. In [3], Moser proved that the integral manifold contains at least one periodic solution whose period is close to that of a periodic solution of the linearized system near the equilibrium point. In [4], Weinstein proved that a Hamiltonian system possesses at least one periodic solution on each energy surface, provided that this energy surface is compact, convex, and contains no stationary point of the vector field.

Bifurcation theory describes how the dynamics of systems change as parameters varied. The study of the Hamiltonian Hopf bifurcation has a long history. The Hamiltonian-Hopf bifurcation involves the loss of linear stability of a fixed point by the collision of two pairs of imaginary eigenvalues of the linearized flow and their subsequent departure off the imaginary axis. van der Meer [5] studied this bifurcation and classified the periodic solutions that are spawned by this resonance. Using 𝑍₂ singularity theory with a distinguished parameter developed in [6], Bridges [7] obtained the periodic solutions in a Hamiltonian-Hopf bifurcation.

Moreover, reversible systems have been studied for many years. Devaney [8] first proved a Liapunov Center theorem for reversible vectors fields. Vanderbauwhede [9] and Sevryuk [10] also studied reversible vectors fields. In [11], Golubitsky et al. studied families of periodic solutions near generic elliptic equilibria for reversible equivariant systems. In [12, 13], Montaldi et al. considered families of periodic solutions near generic elliptic equilibria for reversible equivariant Hamiltonian systems that are both Hamiltonian and reversible at the same time. Since Lamb and Roberts [14] obtained the group theoretical classification of linear reversible equivariant systems, there has been an increasing interest for reversible equivariant systems. Later, Hoveijn et al. [15] obtained the linear normal form and unfolding theory of reversible equivariant linear systems.

In [16], Buzzi and Lamb obtained a Liapunov Center theorem for purely reversible Hamiltonian vector fields that are both Hamiltonian and reversible at the same time. They obtained the existence of periodic solutions in the neighbourhood of elliptic equilibria when the reversing symmetry 𝑅 acts symplectically or antisymplectically. Previously, the symmetric property of periodic solutions is not considered, and the existence of additional periodic solutions is not ruled out. But these problems were considered in [16]. The results in [16] are as follows. If 𝑅 acts antisymplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing a one-parameter family of symmetric periodic solutions. If 𝑅 acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds, each containing a one-parameter family of nonsymmetric periodic solutions, and a three-dimensional invariant manifold containing a two-parameter family of symmetric periodic solutions. In [17], Sternberg theorem for equivariant Hamiltonian vector fields was considered.

Motivated by [16], in this paper, we consider a Liapunov Center theorem for equivariant Hamiltonian vector fields that are both Hamiltonian and equivariant at the same time. Here, we assume that the equivariant symmetry 𝑆 acts antisymplectically and 𝑆²=𝐼. Now, we also consider the symmetric property of periodic solutions. This property was not studied for Hamiltonian vector fields without the other structure previously.

2. Main Results

Theorem 2.1. Consider an equilibrium 0 of a 𝐶^∞ equivariant Hamiltonian vector field 𝑓, with the equivariant symmetry 𝑆 acting antisymplectically and 𝑆²=𝐼. Assume that the Jacobian matrix 𝐷𝑓(0) has two pairs of purely imaginary eigenvalues ±𝑖 and no other eigenvalues of the form ±𝑘𝑖, 𝑘∈𝑍. Then, the equilibrium is contained in a two-dimensional flow-invariant surface that consists of a one-parameter family of symmetric periodic solutions whose period tends to 2𝜋 as they approach the equilibrium. Moreover, the equilibrium is also contained in two smooth two-dimensional flow-invariant manifolds, each containing a one-parameter family of nonsymmetric periodic solutions whose period tends to 2𝜋 as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 2𝜋 in the neighbourhood of 0.

Remark 2.2. Here, the existence and the symmetric property of periodic solutions near the equilibrium point are all considered. The main idea is similar to [16].

3. Linear Equivariant Hamiltonian Vector Field with Purely Imaginary Eigenvalues

We now consider the persistent occurrence of purely imaginary eigenvalues in equivariant Hamiltonian vector fields.

Let 𝐴₀ be a linear Hamiltonian vector field. Then, it follows that 𝐴₀𝐽=−𝐽𝐴^𝑇₀. If 𝐴₀ is 𝑆-equivariant, we have 𝐴₀𝑆=𝑆𝐴₀. If 𝑆 is (anti)symplectic, we get 𝑆𝐽=±𝐽𝑆.

Since we are interested in (partially) elliptic equilibria, we assume that 𝐴₀ has a pair of purely imaginary eigenvalues 𝜆 and −𝜆. Moreover, if the eigenvector 𝑒₁ of 𝐴₀ has 𝜆, then 𝑒₁ is an eigenvector for 𝜆.

Since 𝐴₀ is both Hamiltonian and 𝑆-equivariant, this implies that if the eigenvector 𝑒 of 𝐴₀ has the eigenvalue 𝜆, then 𝑆𝑒 is also an eigenvector for the eigenvalue 𝜆.

Hoveijn et al. [15] considered the linear normal form theory which is based on the construction of minimal ⟨𝐽,𝑆⟩-invariant subspaces. By [15], we are only interested in minimal invariant subspaces on which 𝐴₀ is semisimple; that is, 𝐴₀ is diagonalizable over 𝒞. Here, the type of minimal invariant subspace depends on whether 𝑆 acts symplectically or antisymplectically.

Lemma 3.1. Consider a linear  𝑆-equivariant Hamiltonian vector field  𝐴₀  with  𝑆acting (anti)symplectically and  𝑆²=𝐼. Let 𝑉  be a minimal  (𝐴₀,𝐽,𝑆)-invariant subspace on which  𝐴₀  has purely imaginary eigenvalues. Then  𝐴₀|_𝑉, 𝐽|_𝑉  and  𝑆|_𝑉  have the following normal forms.(1)If 𝑆 acts symplectically, it follows that dim𝑉=2, and 𝑆|_𝑉=|1001,𝐽_𝑉=0−110,𝐴_0𝑉=0−110. (3.1)(2)If 𝑆 acts antisymplectically, it follows that dim𝑉=4 and𝑆|_𝑉=|0010000110000100,𝐽_𝑉=0−1001000000100−10,𝐴_0𝑉=0−1001000000−10010. (3.2)

Proof. Let 𝑊 be a 2-dimensional symplectic subspace on which 𝐴₀ has purely imaginary eigenvalues. By standard Hamiltonian theory and multiplication of time by a scalar, 𝐴₀ and 𝐽 can take the same normal form on 𝑊. If the equivariant symmetry 𝑆 acts symplectically, we have 𝑆𝐴₀=𝐴₀𝑆 and 𝑆𝐽=𝐽𝑆. Let 𝑒₁ and 𝑒₁ be the eigenvectors of 𝐴₀. By 𝑆𝐴₀=𝐴₀𝑆, a minimal invariant subspace is obtained by choosing 𝑆𝑒₁=𝑒₁. Since 𝐽=𝐴₀ on 𝑊, we have 𝑆𝐽=𝐽𝑆 on 𝑊.
If 𝑆 acts antisymplectically, the dimension of the minimal invariant subspace is not two. A 2-dimensional subspace 𝑊 is defined as above. Assume that 𝑆(𝑊)=𝑊. If 𝑆(𝑊)=𝑊, by (𝑆𝐴₀)|_𝑊=(𝐴₀𝑆)|_𝑊, it follows that (𝑆𝐽)|_𝑊=(𝐽𝑆)|_𝑊. This is converse that 𝑆 acts antisymplectically. So, we have 𝑆(𝑊)=𝑊′≠𝑊 and a minimal invariant subspace is given by 𝑉=𝑊⊕𝑊′. So, dim𝑉=4. Moreover, we get 𝐽|_𝑊′=(𝑆⁻¹𝐽𝑆)|_𝑊=−(𝑆⁻¹𝑆𝐽)|_𝑊=−𝐽|_𝑊 and 𝐴₀|_𝑊′=(𝑆⁻¹𝐴₀𝑆)|_𝑊=(𝑆⁻¹𝑆𝐴₀)|_𝑊=𝐴₀|_𝑊. Since 𝐴₀|_𝑊=𝐽|_𝑊, it follows that 𝐽∣_𝑊′=−𝐴₀|_𝑊′.

Remark 3.2. Now, we give the examples for the system (1.1) whether 𝑆 acts symplectically or antisymplectically, where 𝐽 and 𝑆 here are defined as 𝐽|_𝑉 and 𝑆|_𝑉 in Lemma 3.1. If 𝑆 acts symplectically, the system (1.1) can be written as 𝑥̇₁𝑦₁̇=0−1𝐻10𝐻^𝑥1_𝑦1=−𝑦₁−3𝑦²₁𝑥₁+3𝑥²₁=𝑓(𝑥), (3.3) where the Hamiltonian function is 𝐻(𝑥₁,𝑦₁)=(1/2)(𝑥²₁+𝑦²₁)+𝑥³₁+𝑦³₁, 𝑓 satisfies 𝑓𝑆=𝑆𝑓, and 𝐴₀=𝑑𝑓(0) is calculated the same as 𝐴₀|_𝑉 in Lemma 3.1. If 𝑆 acts antisymplectically, the system (1.1) can be written as 𝑥̇₁𝑦₁̇𝑦₂̇𝑥̇₂=0−100𝐻1000000100−10^𝑥1𝐻_𝑦1𝐻_𝑦2𝐻_𝑥2=−𝑦₁−𝑥₁𝑦₂𝑥₁+𝑦₁𝑦₂−𝑥₂𝑦₂−𝑥₂−𝑥₁𝑦₂𝑦₂−𝑥₁𝑦₁+𝑥₁𝑥₂=𝑓(𝑥), (3.4) where the Hamiltonian function is 𝐻(𝑥₁,𝑦₁,𝑦₂,𝑥₂)=(1/2)(𝑥²₁+𝑦²₁)−(1/2)(𝑥²₂+𝑦²₂)+𝑥₁𝑦₁𝑦₂−𝑥₁𝑥₂𝑦₂, 𝑓 satisfies 𝑓𝑆=𝑆𝑓, and 𝐴₀=𝑑𝑓(0) is calculated the same as 𝐴₀|_𝑉 in Lemma 3.1.

Remark 3.3. When 𝑆 acts antisymplectically, under the base (𝑒₁,𝑒₂,𝑆𝑒₁,𝑆𝑒₂), we obtain 𝑆, 𝐽 and 𝐴₀ have the forms of 𝑆|_𝑉, 𝐽|_𝑉 and 𝐴₀|_𝑉 in Lemma 3.1, respectively. However, when 𝑆 acts antisymplectically, under the base (𝑒₁,𝑆𝑒₂,𝑒₂,𝑆𝑒₁), we have 𝑆|_𝑉=|0001001001001000,𝐽_𝑉=00−10000−110000100,𝐴_0𝑉=00−10000110000−100. (3.5) In this case, 𝐽|_𝑉 is the standard form. However, for convenience, we use the forms of Lemma 3.1 in this paper.

4. Liapunov-Schmidt Reduction

In this paper, when 𝑆 acts antisymplectically or symplectically, by Lemma 3.1, 𝐴₀ has a single pair or double pairs of purely imaginary eigenvalues ±𝑖. Moreover, these purely imaginary eigenvalues of 𝐴₀ are nonresonant; that is, 𝐴₀ has no other eigenvalues of the form ±𝑘𝑖 with 𝑘∈𝑍. This condition is clearly generic (codimension zero). We want to find the families of periodic solutions in the neighbourhood of the equilibrium point.

In this section, we introduce the main technique which is a Liapunov-Schmidt reduction. The Liapunov-Schmidt reduction here is similar to the one in [16, 18, 19].

Assume that a 𝐶^∞ vector field 𝑓∶𝑂⊂ℛ^𝑁→ℛ^𝑁 has an equivariant symmetry group 𝐺, which implies the existence of representations 𝜌∶𝐺→𝑂(𝑁) such that 𝑓𝜌(𝛾)=𝜌(𝛾)𝑓, for all 𝛾∈𝐺. Define 𝐹∶𝐶¹_2𝜋×ℛ→𝐶⁰_2𝜋 by𝐹(𝑢,𝜏)=(1+𝜏)𝑑𝑢(𝑢),𝑑𝑠−𝑓 (4.1) where 𝐶¹_2𝜋 is the space of 𝐶¹ maps 𝑢∶𝑆¹→ℛ^𝑁 and 𝐶⁰_2𝜋 is the space of 𝐶⁰ maps 𝑣∶𝑆¹→ℛ^𝑁. The map 𝐹 is 𝐶^∞ by the “Ω-lemma,” that is, in Section  2.4 of [20]. Clearly, the solutions of 𝐹(𝑢,𝜏)=0 correspond to 2𝜋/(1+𝜏)-periodic solutions of (1.1).

Now, define an action 𝑇∶𝐺×𝐶⁰_2𝜋→𝐶⁰_2𝜋 or in 𝐶¹_2𝜋 by𝑇_𝑔𝑢(𝑡)=𝜌𝛾(𝑢(𝑡+𝜃)), (4.2) where 𝑔=𝛾𝜃 is an element of 𝐺, 𝐺=𝐺×𝑆¹, 𝛾∈𝐺 and 𝜃∈𝑆¹.

By the 𝐺-equivariance of 𝑓, we have that 𝐹 is 𝐺-equivariant𝐹𝑇_𝑔𝑢,𝜏=𝑇_𝑔𝐹(𝑢,𝜏),∀𝑔=𝛾𝜃∈𝐺. (4.3)

Assume that 𝑓(0)=0. The derivative of 𝐹 at 𝑢=0 is 𝐿, where𝐿𝑣=𝑑𝐹(0,0)⋅𝑣=𝑣^′−𝐴₀𝑣, (4.4) with 𝐴₀=𝐷𝑓(0). Moreover, ker𝐿=span{Re(𝑒^𝑖𝑠𝑣₀),Im(𝑒^𝑖𝑠𝑣₀)}={Re(𝑧𝑒^𝑖𝑠𝑣₀)∣𝑧∈𝒞}, where 𝐴₀𝑣₀=𝑖𝑣₀.

By (4.3), 𝐿 is also 𝐺-equivariant such that 𝐿𝑇_𝑔=𝑇_𝑔𝐿. Then, 𝑇_𝑔 preserves ker𝐿 and Range 𝐿.

Below, we will obtain that (ker𝐿)^⟂ and (Range𝐿)^⟂ are also 𝑇_𝑔-invariant. We have𝐶¹_2𝜋=ker𝐿⊕(ker𝐿)^⟂,𝐶⁰_2𝜋=Range𝐿⊕Range𝐿^⟂. (4.5) Here, the orthogonal complement is taken in 𝐶⁰_2𝜋 and 𝐶¹_2𝜋 by[𝑢,𝑣]=_𝐺𝑇_𝑔𝑢,𝑇_𝑔𝑣𝑑𝜇, (4.6) where ⟨𝑢,𝑣⟩=∫₀^2𝜋[𝑢(𝑡)]^𝑡𝑣(𝑡)𝑑𝑡 and 𝜇 is a normalized Haar measure for 𝐺. Note that [𝑇_𝑔𝑢,𝑇_𝑔𝑣]=[𝑢,𝑣] for all 𝑔∈𝐺. So, (ker𝐿)^⟂ and (Range𝐿)^⟂ are 𝑇_𝑔-invariant.

By Range𝐿 and (Range𝐿)^⟂ are 𝑇_𝑔-invariant, we can obtain that the projections𝑃∶𝐶⁰_2𝜋⟶Range𝐿,(𝐼−𝑃)∶𝐶⁰_2𝜋⟶Range𝐿^⟂ (4.7) commute with 𝑇_𝑔. If 𝑢∈𝐶⁰_2𝜋, let 𝑢=𝑣+𝑤, where 𝑣∈(Range𝐿)^⟂ and 𝑤∈Range𝐿. Then, 𝑇_𝑔(𝑃(𝑢))=𝑇_𝑔𝑤=𝑃(𝑇_𝑔𝑤)=𝑃(𝑇_𝑔𝑣+𝑇_𝑔𝑤)=𝑃(𝑇_𝑔𝑢) and 𝑇_𝑔((𝐼−𝑃)(𝑢))=𝑇_𝑔𝑣=(𝐼−𝑃)𝑇_𝑔𝑣=(𝐼−𝑃)(𝑇_𝑔𝑣+𝑇_𝑔𝑤)=(𝐼−𝑃)(𝑇_𝑔𝑢).

Next, define a 𝐶^∞ map 𝜔∶ker𝐿×ℛ→(ker𝐿)^⟂ with 𝜔(0,0)=0 by solving𝑃𝐹(𝑘+𝜔,𝜏)=0, (4.8) for 𝜔=𝜔(𝑘,𝜏) using the implicit function theorem.

Moreover, we can prove that 𝜔 commutes with 𝑇_𝑔. Define 𝜔_𝑔∶ker𝐿×ℛ→(ker𝐿)^⟂ by 𝜔_𝑔(𝑘,𝜏)=𝑇_𝑔−1𝜔(𝑇_𝑔𝑘,𝜏). Note that 𝑇_𝑔𝑘∈ker𝐿. We have 𝑃𝐹(𝑘+𝜔_𝑔(𝑘,𝜏),𝜏)=𝑃𝐹(𝑇_𝑔−1(𝑇_𝑔𝑘+𝜔(𝑇_𝑔𝑘,𝜏)),𝜏)=𝑇_𝑔−1𝑃𝐹(𝑇_𝑔𝑘+𝜔(𝑇_𝑔𝑘,𝜏),𝜏)=0. Then, 𝜔_𝑔 is also the solution of (4.8). Moreover, 𝜔_𝑔(0,0)=𝜔(0,0)=0. By uniqueness, we get 𝜔_𝑔(𝑘,𝜏)=𝜔(𝑘,𝜏). So, 𝜔 commutes with 𝑇_𝑔. Then,𝑇_𝑔𝜔(𝑘,𝜏)=𝜔𝑇_𝑔𝑘,𝜏. (4.9)

Then, solutions of the equation 𝐹(𝑢,𝜏)=0 are given by 𝑢=𝑘+𝜔(𝑘,𝜏), where 𝑘 is a solution of the bifurcation equation𝜑(𝑘,𝜏)=(𝐼−𝑃)𝐹(𝑘+𝜔(𝑘,𝜏),𝜏)=0. (4.10)

Now, we will obtain that 𝜑(𝑘,𝜏) is also 𝐺-equivariant.

Lemma 4.1. If 𝑓 is 𝐺-equivariant, then the bifurcation map 𝜑 is 𝐺-equivariant𝜑𝑇_𝑔𝑘,𝜏=𝑇_𝑔𝜑(𝑘,𝜏),∀𝑔∈𝐺×𝑆¹. (4.11)

Proof. Since 𝐺-equivariance of 𝐼−𝑃, 𝐹 and 𝜔(𝑘,𝜏), it is easy to obtain this result.

Lemma 4.1 indicates that how the symmetry enters the bifurcation equation 𝜑. Below, we will also consider the relation between the symmetry and the Hamiltonian function of 𝜑.

Since the vector field 𝑓=𝑋_𝐻 is Hamiltonian, it follows that𝜔𝑋_𝐻(𝑢),𝑣=⟨𝑑𝐻(𝑢),𝑣⟩, (4.12) holds for all 𝑣∈ℛ^2𝑛. Define the mapΦ∶𝐶¹_2𝜋×ℛ×𝐶⁰_2𝜋⟶𝐶¹_2𝜋^∗, (4.13) byΦ(𝑢,𝜏,𝑣)⋅𝑈=₀^2𝜋𝜔𝑣−(1+𝜏)𝑑𝑢(𝑢)𝑑𝑠,𝑈+𝑑𝐻⋅𝑈𝑑𝑠. (4.14) Since the vector field 𝑓 is Hamiltonian with the Hamiltonian function 𝐻, we have that the implicit constraint Φ satisfiesΦ(𝑢,𝜏,𝐹(𝑢,𝜏))=0. (4.15) This condition is rephrased as saying that the map 𝐹, regarded as a (parameter-dependent) vector field on 𝐶¹_2𝜋 and is Hamiltonian with respect to the weak symplectic formΩ()=1𝑢,𝑣2𝜋₀^2𝜋𝜔(𝑢(𝑠),𝑣(𝑠))𝑑𝑠, (4.16) and with the Hamiltonian functionℋ()=1𝑢,𝜏2𝜋₀^2𝜋12𝜔(1+𝜏)𝑑𝑢(𝑢𝑑𝑠,𝑢−𝐻)𝑑𝑠. (4.17)

If the actions of 𝐺 are antisymplectic, we have𝜔𝛾𝑥,𝛾𝑦=−𝜔𝑥,𝑦. (4.18) In this case, Ω is 𝐺=𝐺×𝑆¹ anti-invariant; that is,Ω𝑔𝑢,𝑔𝑣=−Ω(𝑢,𝑣),𝑔=𝛾𝜃,𝛾∈𝐺,𝜃∈𝑆¹, (4.19) and the Hamiltonian function ℋ satisfiesℋ𝑔𝑢,𝜏=−ℋ(𝑢,𝜏). (4.20)

By Theorem  6.2 in [18], it follows that ifker𝐿=ker𝐿^∗, (4.21) then the bifurcation map 𝜑 is also a Hamiltonian vector field with Hamiltonian ℎ(𝑘)=ℋ(𝑘+𝜔(𝑘)), and 𝜑 and the function ℎ have the same invariance properties as the given Hamiltonian ℋ.

In this paper, ker𝐿 is finite dimensional and (4.21) holds. So, the bifurcation equation 𝜑 is a Hamiltonian vector field with Hamiltonian ℎ(𝑘,𝜏)=ℋ(𝑘+𝜔(𝑘),𝜏) and if the actions of 𝐺 are antisymplectic, thenℎ𝑔𝑘,𝜏=−ℎ(𝑘,𝜏). (4.22) The corresponding symplectic form is the restriction of Ω to ker𝐿. Moreover, in this paper, we have 𝐺=𝒵₂ and 𝐺=𝒵₂×𝑆¹≅𝑂(2).

5. Proof of Theorem 2.1

In this section, we prove Theorem 2.1. Theorem 2.1 can be proved by Lemmas 5.1 and 5.3. By Lemma 5.1, the symmetric periodic solutions are obtained. The existence of the nonsymmetric periodic solutions can be verified by Lemma 5.3.

Below, we consider the case that the equivariant symmetry 𝑆 acts antisymplectically. By Lemma 3.1, in this case, there are two pairs of purely imaginary eigenvalues. Without loss of generality, 𝑆|_𝑉, 𝐽|_𝑉 and 𝐴₀|_𝑉 take the (normal) forms of Lemma 3.1, where 𝑉 is the four-dimensional eigenspace of ±𝑖 for 𝐴₀.

Note that dim ker 𝐿=4 and (4.21) holds. By Theorem  6.2 in [18], the bifurcation equation 𝜑 is also a Hamiltonian vector field. We now proceed to apply the Liapunov-Schmidt reduction of Section 4.

Since ker𝐿≅ℛ⁴≅𝒞², it follows thatℛ⁴∋𝑥₁,𝑦₁,𝑥₂,𝑦₂≅𝑥₁+𝑖𝑦₁,𝑥₂+𝑖𝑦₂=𝑧₁,𝑧₂∈𝒞². (5.1) So, the bifurcation equation is denoted by𝜑∶𝒞²×ℛ⟶𝒞². (5.2) Since 𝜑 is Hamiltonian, let 𝜑=2𝐽∇_𝑧ℎ with the Hamiltonian functionℎ∶𝒞²×ℛ⟶ℛ, (5.3) where 𝐽 takes the (normal) form of Lemma 3.1. By (4.22), it follows that ℎ is 𝑆¹-invariant, ℎ∘𝜃=ℎ with 𝜃∈𝑆¹ acting on 𝒞² as𝜃𝑧₁,𝑧₂=𝑒^−𝑖𝜃𝑧₁,𝑒^−𝑖𝜃𝑧₂, (5.4) and ℎ is 𝑆-anti-invariant, ℎ∘𝑆=−ℎ with𝑆𝑧₁,𝑧₂=𝑧₂,𝑧₁. (5.5)

By 𝑆¹-invariance, it follows thatℎ=ℎ𝑧₁²,𝑧₂²,𝑧₁𝑧₂,𝑧₂𝑧₁,𝜏. (5.6)

Then, by the 𝑆-anti-invariance, we have thatℎ=𝑧₁²−𝑧₂²𝜓𝑧₁²+𝑧₂²,𝑧₁𝑧₂²,𝑧₁𝑧₂,𝑧₂𝑧₁,𝜏 (5.7) is equivalent to (𝜕^𝑛𝜓/𝜕(𝑧₁𝑧₂)^𝑛)|_𝑧1𝑧2=0 is real for all 𝑛,𝑧₁ and 𝑧₂ which satisfy 𝑛∈𝒵 and 𝑧₁𝑧₂=0. If (𝜕^𝑛𝜓/𝜕(𝑧₁𝑧₂)^𝑛)|_𝑧1𝑧2=0 is not real for some 𝑛,𝑧₁ and 𝑧₂ which satisfy 𝑛∈𝒵 and 𝑧₁𝑧₂=0, by the 𝑆-anti-invariance, we have ℎ=(|𝑧₁|²−|𝑧₂|²)𝜓(|𝑧₁|²+|𝑧₂|²,|𝑧₁𝑧₂|²,𝜏). Thus, without generality, since ℎ is real, we obtainℎ=𝑧₁²−𝑧₂²𝜓𝑧₁²+𝑧₂²,𝑧₁𝑧₂²,𝑧₁𝑧₂,𝑧₂𝑧₁=𝑧,𝜏₁₂−𝑧₂²𝜓¹𝑧₁²+𝑧₂²,𝑧₁𝑧₂²𝑧,𝜏+2Re₁𝑧₂𝜓²𝑧₁²+𝑧₂²,𝑧₁𝑧₂²,𝑧₁𝑧₂,𝜏, (5.8) where 𝜓=𝜓¹+2Re(𝑧₁𝑧₂𝜓²).

We have that ker𝐿 is generated by𝑣₁=(sin𝑠,−cos𝑠,0,0)^𝑇,𝑣₂=(cos𝑠,sin𝑠,0,0)^𝑇,𝑣₃=(0,0,sin𝑠,−cos𝑠)^𝑇,𝑣₄=(0,0,cos𝑠,sin𝑠)^𝑇. (5.9)

Then, the symplectic form Ω for the reduced bifurcation equation satisfiesΩ𝑣_𝑛,𝑣_𝑚=12𝜋₀^2𝜋𝑣_𝑛,𝐽𝑣_𝑚𝑑𝑠, (5.10) where 𝐽 takes the (normal) form of Lemma 3.1. So,Ω𝑘₁,𝑘₂=𝑘₁,𝐽𝑘₂, (5.11) for all 𝑘₁,𝑘₂∈ker𝐿.

Using the similar calculation and by (4.17), it follows that the 𝜏-dependence of the lowest (quadratic) order of ℎ has the formℎ=𝑧₁²−𝑧₂²𝜏2+𝑙𝑧₁,𝑧₂,𝜏. (5.12)

Using (5.8) and (5.12), we have 𝜓¹(0,0,𝜏)=𝜏/2.

We first give Lemma 5.1, which is used for finding the symmetric periodic orbits near the equilibrium point.

5.1. Symmetric Periodic Solutions

Since ℎ∘𝑆=−ℎ, then the symmetric periodic solutions of the bifurcation equation lie in the level set ℎ=0. In fact, if ℎ=0, we get 𝜑=0. Then, by (4.9), the symmetric solutions lying in ℎ=0 correspond to symmetric 2𝜋-periodic solutions for (4.1). Below, we prove there exist the symmetric solutions for the equation ℎ=0.

Lemma 5.1. In the neighbourhood of the equilibrium (𝑧₁,𝑧₂,𝜏)=(0,0,0), there exist the symmetric solutions for the equation ℎ=0.

Proof. Since 𝑆(𝑧₁,𝑧₂)=(𝑧₂,𝑧₁), the symmetric solutions lie in Fix(𝑆)={(𝑧,𝑧)∈𝒞²}. Then, the equation 𝜑=2𝐽∇_𝑧ℎ=0 is equivalent to 𝑧=0 or 𝜓(𝑧,𝜏)=0. By (5.8) and (5.12), we have (𝜕𝜓/𝜕𝜏)(0,0)=1/2≠0. Using the implicit function theorem for 𝜓=0, there exists a function 𝜏=𝜏(|𝑧|²) for all sufficiently small |𝑧| that corresponds to a family of symmetric periodic solutions of the system (1.1) with period 2𝜋/(1+𝜏).

Remark 5.2. Apparently, the symmetric periodic solutions should be twoparameters. But since the vector field 𝑓 is the equivariant Hamiltonian vector field, then the symmetric periodic solutions are in fact one-parameter.
Next, we study the nonsymmetric periodic solutions in the neighbourhood of the equilibrium 0.

5.2. Nonsymmetric Periodic Solutions

Lemma 5.3. Except the symmetric Liapunov Center family described in Lemma 5.1, there are two nonsymmetric Liapunov Center families of periodic solutions, each contained in a local two-dimensional smooth manifold with the period of the periodic solutions converging to 2𝜋 as the solutions tend to the equilibrium point.

Proof. The proof can be divided into three cases: 𝑧₁≠0,𝑧₂≠0𝑎𝑛𝑑𝑧₁≠𝑧₂,𝑧₁=0and𝑧₂≠0, and 𝑧₁≠0and𝑧₂=0.
𝑧₁≠0,𝑧₂≠0and𝑧₁≠𝑧₂: let 𝐼₁=|𝑧₁|²−|𝑧₂|². It follows that 𝜑=0 is equivalent to ∇_𝑧ℎ=0. By (5.8), ∇_𝑧ℎ=0 can be written as 𝑧₁−𝑧₂𝜓+𝐼₁∇_𝑧𝜓=0. (5.13)
By (5.8), and multiplying the first equation of (5.13) by 𝑧₁ and the second equation of (5.13) by 𝑧₂, we have𝑧₁²𝜓+𝐼₁𝜓¹₁+𝑧₂²𝜓¹₂+2Re𝑧₁𝑧₂𝜓²₁+𝑧₂²𝜓²₂+2𝐼₁𝑧₂𝑧₁𝜓²+𝑧₁𝑧₂𝜓²₃𝑧=0,₂₂−𝜓+𝐼₁𝜓¹₁+𝑧₁²𝜓¹₂+2Re𝑧₁𝑧₂𝜓²₁+𝑧₁²𝜓²₂+2𝐼₁Re𝑧₁𝑧₂𝜓²+𝑧₁𝑧₂𝜓²₃=0, (5.14)(5.15) where 𝜓^𝑗_𝑖 is the partial derivative of 𝜓^𝑗 with respect to 𝑋_𝑖, 𝑋₁=|𝑧₁|²+|𝑧₂|², 𝑋₂=|𝑧₁𝑧₂|², and 𝑋₃=𝑧₁𝑧₂.
Now, we consider the real parts of (5.14) and (5.15). Subtracting the real part of (5.14) from (5.15), we obtain 𝑋₁𝜓+𝐼²₁𝜓¹₁+2𝐼²₁Re𝑧₁𝑧₂𝜓²₁=0. (5.16) So, 𝑋₁𝜓+𝐼²₁𝜕𝜓𝜕𝑋₁=0. (5.17) By (5.17), we have 𝜓=𝑐𝑒^{−𝑋21/2𝐼21}, (5.18) where 𝑐 is a constant. Let us take (5.18) into ℎ=𝐼₁𝜓 and verify ∇_𝑧ℎ=0. Then, the first equation ∇_𝑧1ℎ=0 becomes 𝑐𝑒^{−𝑋21/2𝐼21}𝑧₁−𝐼₁𝑐𝑒^{−𝑋21/2𝐼21}⋅4𝑋₁𝑧₁𝐼²₁−4𝐼₁𝑧₁𝑋²₁4𝐼⁴₁=0. (5.19) If 𝜓≠0, it follows that 𝐼1=¹𝑋₁−𝑋²₁𝐼²₁. (5.20) By (5.20), we get 𝑧₁⁴+𝑧₂⁴=−2𝑧₂⁴. (5.21) But (5.21) does not hold in this case. Therefore, we find no small solutions of the bifurcation equation for 𝑧₁≠0, 𝑧₂≠0and 𝑧₁≠𝑧₂.
𝑧₁=0and𝑧₂≠0: in this case, the equation ∇_𝑧ℎ=0 is equivalent to 𝑧₂𝜓¹+𝑧₂²𝜓¹₁=0⟺𝑟𝑧₂²,𝜏=𝜓¹+𝑧₂²𝜓¹₁=0. (5.22) Moreover, (𝜕𝑟/𝜕𝜏)|_{(|𝑧2|2,𝜏)=(0,0)}=1/2. Using the implicit function theorem for (5.22), there exists a function 𝜏=𝜏(|𝑧₂|²) with 𝜏(0)=0 for all sufficiently small |𝑧₂|. Correspondingly, we have a one-parameter family of nonsymmetric periodic solutions of the system (1.1) contained in a local smooth two-dimensional invariant manifold.
𝑧₁≠0and𝑧₂=0: similarly to the above case, for 𝜑=0, there exists a function 𝜏=𝜏(|𝑧₁|²) with 𝜏(0)=0 for all sufficiently small |𝑧₁|. Correspondingly, there is another one-parameter family of nonsymmetric periodic solutions of the system (1.1) filling out a local smooth two-dimensional invariant manifold. This family of nonsymmetric periodic solutions are the 𝑆-image of the family with 𝑧₁=0 and 𝑧₂≠0.

Remark 5.4. We now consider the case that 𝑆 acts symplectically. In this case, purely imaginary eigenvalues pairs typically arise isolated. Since 𝑆 on the two-dimensional eigenspace 𝑉 of ±𝑖 for 𝐴₀ is the identity matrix as in Lemma 3.1, then the symmetric property of the periodic solutions is insignificant. However, we can prove the existence of a Liapunov Center family using the fact that the flow is Hamiltonian [18, 21].

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions.